Partitioning Data into Clusters

Cluster analysis is an unsupervised learning technique used for classification of data. Data elements are partitioned into groups called clusters that represent proximate collections of data elements based on a distance or dissimilarity function. Identical element pairs have zero distance or dissimilarity, and all others have positive distance or dissimilarity.

The rule-based data syntax can also be used to cluster data based on parts of each data entry. For instance, you might want to cluster data in a data table while ignoring particular columns in the table.

In principle, it is possible to cluster points given in an arbitrary number of dimensions. However, it is difficult at best to visualize the clusters above two or three dimensions. To compare optional methods in this documentation, an easily visualizable set of two-dimensional data will be used.

The following commands define a set of 300 two-dimensional data points chosen to group into four somewhat nebulous clusters.

Randomness is used in clustering in two different ways. Some of the methods use a random assignment of some points to a specific number of clusters as a starting point. Randomness may also be used to help determine what seems to be the best number of clusters to use. Changing the random seed for generating the randomness by using FindClusters[{e1,e2,…},Method->{Automatic,"RandomSeed"->s}] may lead to different results for some cases.

In principle, clustering techniques can be applied to any set of data. All that is needed is a measure of how far apart each element in the set is from other elements, that is, a function giving the distance between elements.

FindClusters[{e1,e2,…},DistanceFunction->f] treats pairs of elements as being less similar when their distances are larger. The function f can be any appropriate distance or dissimilarity function. A dissimilarity function satisfies the following:

If the are vectors of numbers, FindClusters by default uses a squared Euclidean distance. If the are lists of Boolean True and False (or 0 and 1) elements, FindClusters by default uses a dissimilarity based on the normalized fraction of elements that disagree. If the are strings, FindClusters by default uses a distance function based on the number of point changes needed to get from one string to another.

Dissimilarities for Boolean vectors are typically calculated by comparing the elements of two Boolean vectors and pairwise. It is convenient to summarize each dissimilarity function in terms of , where is the number of corresponding pairs of elements in and , respectively, equal to and . The number counts the pairs in , with and being either 0 or 1. If the Boolean values are True and False, True is equivalent to 1 and False is equivalent to 0.

The edit distance is determined by counting the number of deletions, insertions, and substitutions required to transform one string into another while preserving the ordering of characters. In contrast, the Damerau–Levenshtein distance counts the number of deletions, insertions, substitutions, and transpositions, while the Hamming distance counts only the number of substitutions.

The methods and determine how to cluster the data for a particular number of clusters k. uses an agglomerative hierarchical method starting with each member of the set in a cluster of its own and fusing nearest clusters until there are k remaining. starts by building a set of k representative objects and clustering around those, iterating until a (locally) optimal clustering is found. The default method is based on partitioning around medoids.

Additional Method suboptions are available to allow for more control over the clustering. Available suboptions depend on the Method chosen.

"SignificanceTest"

test for identifying the best number of clusters

Suboption for all methods.

For a given set of data and distance function, the choice of the best number of clusters k may be unclear. With Method->{methodname,"SignificanceTest"->"stest"}, is used to determine statistically significant clusters to help choose an appropriate number. Possible values of are and . The test uses the silhouette statistic to test how well the data is clustered. The test uses the gap statistic to determine how well the data is clustered.

The test subdivides the data into successively more clusters looking for the first minimum of the silhouette statistic.

The test compares the dispersion of clusters generated from the data to that derived from a sample of null hypothesis sets. The null hypothesis sets are uniformly randomly distributed data in the box defined by the principal components of the input data. The method takes two suboptions: and . The suboption sets the number of null hypothesis sets to compare with the input data. The option sets the sensitivity. Typically larger values of will favor fewer clusters being chosen. The default settings are and .

Note that the clusters found in these two examples are identical. The only difference is how the number of clusters is chosen.

"Linkage"

the clustering linkage to use

Suboption for the method.

With Method->{"Agglomerate","Linkage"->f}, the specified linkage function f is used for agglomerative clustering.

"Single"

smallest intercluster dissimilarity

"Average"

average intercluster dissimilarity

"Complete"

largest intercluster dissimilarity

"WeightedAverage"

weighted average intercluster dissimilarity

"Centroid"

distance from cluster centroids

"Median"

distance from cluster medians

"Ward"

Ward's minimum variance dissimilarity

f

a pure function

Possible values for the suboption.

Linkage methods determine this intercluster dissimilarity, or fusion level, given the dissimilarities between member elements.

With Linkage->f, f is a pure function that defines the linkage algorithm. Distances or dissimilarities between clusters are determined recursively using information about the distances or dissimilarities between unmerged clusters to determine the distances or dissimilarities for the newly merged cluster. The function f defines a distance from a cluster k to the new cluster formed by fusing clusters i and j. The arguments supplied to f are , , , , , and , where d is the distance between clusters and n is the number of elements in a cluster.

These are the clusters found using complete linkage hierarchical clustering.